New PDF release: A First Course in Geometric Topology and Differential

The individuality of this article in combining geometric topology and differential geometry lies in its unifying thread: the idea of a floor. With a variety of illustrations, workouts and examples, the scholar involves comprehend the connection among sleek axiomatic procedure and geometric instinct. The textual content is stored at a concrete point, 'motivational' in nature, keeping off abstractions. a couple of intuitively attractive definitions and theorems relating surfaces within the topological, polyhedral, and delicate circumstances are offered from the geometric view, and element set topology is particular to subsets of Euclidean areas. The remedy of differential geometry is classical, facing surfaces in R3 . the cloth here's available to math majors on the junior/senior point.

Sigurdur Helgason's Differential Geometry and Symmetric areas was once fast famous as a extraordinary and significant publication. for a few years, it was once the normal textual content either for Riemannian geometry and for the research and geometry of symmetric areas. numerous generations of mathematicians depended on it for its readability and cautious realization to element.

The purpose of those lecture notes is to offer an primarily self-contained advent to the fundamental regularity conception for power minimizing maps, together with contemporary advancements about the constitution of the singular set and asymptotics on method of the singular set. really good wisdom in partial differential equations or the geometric calculus of diversifications is no longer required.

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There were many tremendous advancements within the idea of minimum surfaces and geometric degree idea long ago 25 to 30 years. some of the researchers who've produced those first-class effects have been encouraged by way of this little e-book - or by way of Fred Almgren himself. The e-book is certainly a pleasant invitation to the realm of variational geometry.

Extra info for A First Course in Geometric Topology and Differential Geometry

Example text

A Schottky group of rank p is a subgroup of G which has a collection of non-elliptic, non-trivial generators {g1 , . . , gp } satisfying the following condition: there exists a point 0 in D such that the closures in D = D ∪ D(∞) of the sets D0 (gi±1 ) = D(gi±1 ), for i = 1, . . , p, satisfy (D(gi ) ∪ D(gi−1 )) ∩ (D(gj ) ∪ D(gj−1 )) = ∅, for all i = j in {1, . . , p}. Let S(g1 , . . , gp ) denote such a group whose collection of generators is {g1 , . . , gp }. In the rest of this discussion, in order to avoid notational clutter, we will restrict ourselves to the case where p = 2.

16. The ﬁrst two surfaces are topologically Fig. 25. equivalent. Both are homeomorphic to a cylinder. From the metric point of view, however, there are some diﬀerences. To highlight these diﬀerences, choose a generator g of Γ . Deﬁne the curve c = [g −1 (i), i]h ∪ [i, g(i)]h . Let c be the intersection of c with the Dirichlet domain Di (Γ ). Now (referring to Fig. 25) remove c from Di (Γ ). In the ﬁrst case, the curve c is the segment [2i, 1/2i]h , thus c splits the domain into two subdomains of inﬁnite area.

Fig. 20. h(z) = 2z Fig. 21. , for any z and z in Hz0 (γ), the hyperbolic segment [z, z ]h is included in this set). It follows that the intersection of all Hz0 (γ) is also convex. Deﬁne Hz0 (γ). Dz0 (Γ ) = γ∈Γ γ=Id This set is called the Dirichlet domain of Γ centered at z0 . 11. A Dirichlet domain is a convex fundamental domain of Γ . 12. 11. ) Let us examine the boundary of the Dirichlet domain Dz0 (Γ ) in H. 13. The set Dz0 (Γ )− Dz0 (Γ ) is in the union of the perpendicular bisectors of the segments [z0 , γ(z0 )]h , with γ in Γ − {Id}.